Finite element analysis of a nonlinear heat Equation with damping and pumping effects Article Swipe

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Wasim Akram , Manil T. Mohan · YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2510.10210

We study the following nonlinear heat equation with damping and pumping effects (a reaction-diffusion equation) posed on a bounded simply connected convex domain $Ω\subset \mathbb{R}^d$, $d \geq 1$ with Lipschitz boundary $\partialΩ$: $$ \frac{\partial u(t)}{\partial t} - νΔu(t) + α|u(t)|^{p-2}u(t) - \sum_{\ell=1}^M β_{\ell} |u(t)|^{q_{\ell}-2}u(t) = f(t), \quad t>0, $$ subject to homogeneous Dirichlet boundary conditions and the initial condition $u(0)=u_0$, where $2 \leq p < \infty$ and $2 \leq q_{\ell} < p$ for $1 \leq \ell \leq M$. For $u_0 \in L^2(Ω)$ and $f \in L^2(0,T;H^{-1}(Ω))$, we establish the existence and uniqueness of a weak solution for all dimensions $d \in \mathbb{N}$ and damping exponents $2 \leq p < \infty$. Furthermore, for $u_0 \in H^2(Ω) \cap H_0^1(Ω)$ and $f \in H^1(0,T;H^1(Ω))$, we obtain regularity results: these hold for every $2 \leq p < \infty$ when $1 \leq d \leq 4$, and for $2 \leq p \leq \frac{2d-6}{d-4}$ when $d \geq 5$. We further conduct finite element analysis using conforming, nonconforming, and discontinuous Galerkin methods, deriving a priori error estimates for both semi- and fully discrete schemes, supported by numerical results. To relax restrictions on $p$ in the semidiscrete analysis, we use appropriate projection/interpolation operators: the Ritz projection in the conforming case ($2 \le p \le \frac{2d}{d-2}$), the Scott-Zhang interpolation for $\frac{2d}{d-2} < p \le \frac{2d-6}{d-4}$, the Clément interpolation in the nonconforming setting, and the $L^2$-projection in the DG framework. In the fully discrete case, error estimates hold for the above $p$-range under $u_0 \in D(A^{3/2})$ and $f \in H^1(0,T;H^1(Ω))$.

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preprint

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en

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http://arxiv.org/abs/2510.10210

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https://arxiv.org/pdf/2510.10210

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https://openalex.org/W4417100135

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https://openalex.org/W4417100135

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https://doi.org/10.48550/arxiv.2510.10210

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Finite element analysis of a nonlinear heat Equation with damping and pumping effects

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preprint

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en

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2025

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2025-10-11

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Wasim Akram, Manil T. Mohan

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https://arxiv.org/abs/2510.10210

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